Efficient and accurate stress recovery procedure and a posteriori error estimator for the stable generalized/extended finite element method

Abstract

SummaryThis paper presents a new stress recovery technique for the generalized/extended finite element method (G/XFEM) and for the stable generalized FEM (SGFEM). The recovery procedure is based on a locally weighted L2 projection of raw stresses over element patches; the set of elements sharing a node. Such projection leads to a block‐diagonal system of equations for the recovered stresses. The recovery procedure can be used with GFEM and SGFEM approximations based on any choice of elements and enrichment functions. Here, the focus is on low‐order 2D approximations for linear elastic fracture problems. A procedure for computing recovered stresses at re‐entrant corners of any internal angle is also presented. The proposed stress recovery technique is used to define a Zienkiewicz‐Zhu (ZZ) a posteriori error estimator for the G/XFEM and the SGFEM. The accuracy, computational cost, and convergence rate of recovered stresses together with the quality of the ZZ estimator, including its effectivity index, are demonstrated in problems with smooth and singular solutions.